Skip to main content

Questions tagged [complex-manifolds]

For questions about complex manifolds.

Filter by
Sorted by
Tagged with
2 votes
1 answer
72 views

Sheaves of sections of vector bundles

If I have two sheaves $E$ and $G$ over a complex manifold $M$ and I want to prove something like say $\mathcal{O}(E) \otimes_{\mathcal{O}} \mathcal{O}(G) \cong \mathcal{O}(E\otimes G)$, where $\...
Tim's user avatar
  • 207
2 votes
1 answer
60 views

Jet bundle question

Let $E \to M$ be a holomorphic vector bundle. Is there a metric on the first jet bundle $J^1 E$ that can be defined in terms of metrics on $E$ and $M$?
Hammerhead's user avatar
0 votes
1 answer
70 views

Classifying space of finite group as a complex manifold

Suppose $G$ be a finite group. Then, is it always possible to construct a classifying space which is a finite dimensional(if not, possibly infinite dimensional) complex manifold? More precisely, are ...
ChoMedit's user avatar
  • 786
0 votes
0 answers
27 views

Codomain of holomorphic chart

I am beginning to study complex manifolds and have come across the following possible issue when extending from the real to the complex case. Beginning from the real case, we can define a real (smooth)...
Nick F's user avatar
  • 1,249
0 votes
0 answers
31 views

Inverse of pullback metric

Suppose we have the inclusion $\iota: X \hookrightarrow \mathbb{P}^n$ which is injective, but not a diffeomorphism. Given the standard metric $g_{\mu \overline{\nu}} dz^{\mu} \otimes d\overline{z}^{\...
Eweler's user avatar
  • 701
1 vote
0 answers
78 views

Examples of Complex Manifolds

I'm trying to learn about Complex Manifolds, and something that has proven a stumbling block is the paltry number of examples provided. The best of the book I found was Huybrechts Complex Geometry ...
Derivative's user avatar
  • 1,853
1 vote
0 answers
44 views

Foliation by complex hypersurfaces

I am learning about the Levi flat hypersurfaces. Let $M \subseteq \mathbb{C}^n$ be a real smooth hypersurface, i.e. for every point $p$ in $M$ there is an open set $U_p$ in $\mathbb{C}^n$ containing $...
Curious's user avatar
  • 973
0 votes
1 answer
33 views

Endomorphisms of Lie group acting on cotangent space

Let $G$ be a (complex, compact, commutative) Lie group. Apparently the endomorphism ring $\textrm{End}(G)$ of $G$ (i.e., holomorphic group homomorphisms) acts on the cotangent space $T^\ast_eG$ at the ...
Joseph Harrison's user avatar
2 votes
1 answer
101 views

Why is the complex Lie group $(\mathbb C^*)^n$ called "Complex Torus"

While studying complex Lie groups theory, and more generally complex geometry, I've found two different objects which are called "complex tori". Consider the multiplicative group $\mathbb C^...
Federico T.'s user avatar
  • 1,048
2 votes
0 answers
32 views

Why is the stabilizer of a holomorphic complex Lie group action a complex Lie subgroup?

Suppose $G$ is a complex Lie group and $M$ is a complex manifold. Suppose we have an action of $G$ on $M$ which is a holomorphic map $G \times M \rightarrow M$. I have seen the claim that it is easy ...
rosecabbage's user avatar
  • 1,697
3 votes
1 answer
168 views

$\partial_{z}$ and $\partial_{\bar{z}}$: what are these vector fields from a geometrical point of view?

In complex analysis, we are taught that instead of coordinates $x$, $y$ on the complex plane, one can use $z$, $\bar{z}$, then, for instance, the Cauchy-Riemann conditions become $\frac{\partial }{\...
Daigaku no Baku's user avatar
1 vote
1 answer
193 views

Compute first Chern class for complex Torus

I have developed a keen interest in understanding the Calabi-Yau manifold. I have been following "Lectures on Kähler Geometry" by Andrei Moroianu and several online resources. However, it ...
N00BMaster's user avatar
1 vote
1 answer
140 views

Understanding the almost complex structure of a complex manifold

I start learning complex manifold by myself and hard to lift my previous intuition of differential geometry over the complex structure. Let $M$ be a real $2m$-dimensional manifold. We define an ...
N00BMaster's user avatar
0 votes
0 answers
38 views

Is a manifold defined by a complex-valued analytic function on $\mathbb{R}^d$ considered a real-analytic manifold?

Suppose $f:\mathbb{C}^d \to \mathbb{C}$ is a holomorphic function such that ${\partial f \over \partial z_k} \neq 0$ for $k = 1, \ldots, d$ whenever $f(z) = 0$. Suppose also that the zero set of $f$ ...
Clyde's user avatar
  • 903
0 votes
0 answers
81 views

What's the formula for the given code? Calabi Yau manifolds

I need help finding a reference online for the formula that I tried to implement a few years ago and can't find the link anymore. Here's the code. MyFunction return the position of each sphere at any ...
Paulo Renan's user avatar
0 votes
0 answers
112 views

Show that $\frac{1}{2\pi i}\int_{\gamma_p}z\frac{h'(z)}{h(z)}dz$ is in $\mathbb{Z}+\mathbb{Z}\tau$

This is the problem IV.3.F I found at page 127 of the book "Algebraic Curves and Riemann Surfaces" of Rick Miranda. Let $\tau \in \mathbb{C}$ such that $Im(\tau)>0$ and define the lattice ...
100nanoFarad's user avatar
2 votes
0 answers
34 views

Is a module over a (c-)soft sheaf (c-)soft?

In the context of Riemann Surfaces, or $\mathbb C$ manifolds (or even more generally) I wonder if a module over a c-soft sheaf $\mathcal S$ is itself c-soft which I believe implies soft if the space ...
raisinsec's user avatar
  • 463
1 vote
0 answers
56 views

Möbius transformation taking line segment between two complex numbers to the segment $[-1,1]$

I am reading Kodaira's Complex Analysis, and trying to understand his proof of the uniformization theorem for simply connected Riemann surfaces. Due to what I assume is translation errors, this book (...
John Cavanaugh's user avatar
1 vote
0 answers
27 views

Real Description of a Kähler Manifold [duplicate]

If $(M,\omega)$ is a Kähler manifold, then we have the following structures on $M:$ A smooth manifold structure on $M$. An almost complex structure $J.$ A Riemannian metric $g,$ satisfying $$g(X,Y)=g(...
Stewan's user avatar
  • 493
1 vote
0 answers
19 views

Why is the set of points in $\mathbb{C}P^2$ where a non-degenerated quadratic form vanishes biholomorphic to $\mathbb{C}P^1$?

Let $Q: \mathbb{C}^3 \rightarrow \mathbb{C}$ a non-degenerated quadratic form and let $S=\{[z_1,z_2,z_3] \in \mathbb{C}P^2 : Q(z_1, z_2, z_3) \}$. My lecture notes on Riemann surfaces mention that $S$ ...
Gokimo's user avatar
  • 355
2 votes
1 answer
73 views

Prove that $H^0(M,TM) \cong H^0(M_1,TM_1) \oplus H^0(M_2,TM_2).$ Roughly every vector field on $M$ is uniquely the sum of vector fields.

For a complex manifold $N$, denote by $H^0(N,TN)$ the space of holomorphic vector fields on $N$. Let $M_i, i=1,2$ be compact complex manifolds, $M=M_1\times M_2$ their product. Prove that $$H^0(M,TM) \...
Nathaniel Johonson's user avatar
1 vote
0 answers
78 views

Global sections of $\mathcal{O}(k)$ over $\mathbb{P}^1$.

I have the following example from Well's book on differential analysis on complex manifolds. I don't currently understand why they get the description for $\mathcal{O}(\Bbb P_1(\Bbb C), E^k)$ from ...
Nathaniel Johonson's user avatar
4 votes
1 answer
141 views

Nice coordinate presentation of the covering map $S^2 \to \mathbb{R}P^2$

I would like to write down a nice coordinate presentation of the double cover map $S^2 \to \mathbb{R}P^2$, where $S^2$ carries a single complex coordinate $z$ as the Riemann sphere $\mathbb{C}P^1 \...
ziggurism's user avatar
  • 16.9k
4 votes
1 answer
175 views

Tangent spaces of $\Bbb P^1$

Consider the following descriptions of the complex projective line $\Bbb P^1$: The unit sphere $\{(u,v,w) \in \mathbb{R}^3 \mid u^2+v^2+w^2=1\}$ which is identified with the Riemann sphere $\Bbb C \...
Nathaniel Johonson's user avatar
2 votes
1 answer
59 views

Compact connected complex/holomorphic manifold that embeds in $\mathbb{C}^n$

If $i : M \to \mathbb{C}^n$ is a holomorphic map, then each coordinate function on $\mathbb{C}^n$ restricts to a global holomorphic function on the image. In particular, there is no holomorphic ...
Michele's user avatar
  • 35
2 votes
0 answers
66 views

Analytic variety induced by an irreducible polynomial

I am reading Principles of Algebraic Geometry by Griffith and Harris. Here the authors define an analytic variety in a domain as follows : A subset $V$ of an open set $U \subset \mathbb{C}^n$ is an ...
Curious's user avatar
  • 973
0 votes
0 answers
62 views

Holomorphic maps of complex manifolds preserve the bidegree of a complex differential form

For a holomorphic map $f : M \to N$ between complex manifolds show that if $\omega$ is a form of type $(p,q)$ on $N$, then $f^*\omega$ is a form of type $(p,q)$ on $M$. I am wondering if the ...
Sehat's user avatar
  • 311
3 votes
1 answer
93 views

Redundancy with basis vectors for $T_pM^\mathbb C \cong T_pM^{1,0} \oplus T_pM^{0,1}$

I've found out that the primary motivation for complexifying the tangent bundle is to enable its decomposition into the eigenspaces $\pm i$ of $J$. This decomposition facilitates the splitting of the ...
Victor's user avatar
  • 223
2 votes
1 answer
85 views

Basis for $T_pM^\mathbb{C}$

For a real vector space $V$ with dimension $n$ and a basis $\{e_1,\dots,e_n\}$ the complexification $V^\mathbb{C}=V \otimes \mathbb C$ contains of vectors of the form $v+iw$ for $v,w\in V$. Now write $...
Victor's user avatar
  • 223
0 votes
0 answers
52 views

Show that any non-trivial homogeneous polynomial $s$ of degree $k$ can be considered as a non-trivial section of $\mathcal{O}(k)$ on $\mathbb{P}^n$.

Show that any non-trivial homogeneous polynomial $0 \ne s \in \mathbb{C}[z_0, \dots, z_n]$ of degree $k$ can be considered as a non-trivial section of $\mathcal{O}(k)$ on $\mathbb{P}^n$. So $\mathcal{...
Sehat's user avatar
  • 311
1 vote
1 answer
265 views

Structure on a complex manifold and the Kähler form on $\mathbb{C}^2$.

Currently learning about complex and Kähler manifolds and I'm reading this article about Kähler forms. Specifically I'm trying to understand how do they conclude that on $\mathbb{C}^2$ the Kähler form ...
Louie's user avatar
  • 561
0 votes
1 answer
133 views

"Degree function" of holomorphic map between compact Riemann surfaces

I am currently reading the proof about the degree of a holomorphic (nonconstant) function between compact Riemann surfaces in "Algebraic Curves and Riemann Surfaces" by R. Miranda. The ...
Florian Manzini's user avatar
3 votes
0 answers
49 views

Proof of the fact that $\mathfrak{g}\otimes C^{\infty}(S^1) \cong C^{\infty}(S^1;\mathfrak{g})$

I encountered the following fact: let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, and consider the tensor product $\mathfrak{g}\otimes_{\mathbb{C}} C^{\infty}(S^1)$ (where I guess we mean maps ...
toyr99's user avatar
  • 191
0 votes
1 answer
292 views

Proof of Spherical metric on Riemann Sphere

While studying stereographic projection of extended complex on unit sphere $S$ in $\mathbb{R^3}$ we get two metrics one is chordal metric and second one is spherical metric. The spherical metric $d_s$ ...
Nirmal Rawat's user avatar
2 votes
1 answer
117 views

What is the definition of sheaf of meromorphic differential form on a complex manifold?

Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is the sheaf of $\mathbb{C}$-valued holomorphic functions on $X$. Let $T_X^\vee$ be the cotangent bundle over $X$ with the ...
Z Wu's user avatar
  • 1,785
0 votes
1 answer
69 views

$f:M \rightarrow N$ holomorphic between equidimensional complex manifolds is surjective if $|J(f)| \not\equiv 0$

In the book "Principles of algebraic geometry" by Griffiths and Harris (PG. 237) there is a proof of the following statement: "Let $f:M \rightarrow N$ be a holomorphic map between two ...
Tazz's user avatar
  • 3
1 vote
1 answer
121 views

Partial derivatives chart-dependent?

Let $X$ be a Riemann surface. Let $Y\subset X$ be open. The following definitions are taken from page 60 of Forster's Riemann Surfaces. We call a function $f\colon Y \rightarrow \mathbb{C}$ (...
Peter's user avatar
  • 881
1 vote
1 answer
173 views

Singular irreducible affine plane curve is never a Riemann surface?

Let $f\in \mathbb{C}[x,z]$ be a bivariate irreducible polynomial over the complex numbers. In case that $f$ is non-singular, one can endow the locus (zero set) of $f$ (considered with subspace ...
Peter's user avatar
  • 881
1 vote
1 answer
158 views

Multiplicity of a non-constant holomorphic map

In Miranda's Algebraic Curves and Riemann Surfaces he defines the multiplicity of a non-constant holomorphic map $F\colon X \rightarrow Y$ between Riemann surfaces as the unique integer $m$ such that ...
Peter's user avatar
  • 881
0 votes
1 answer
136 views

Holomorphic forms are closed on compact manifold $X$ if $\dim(X)=2$.

Let $X$ be a compact complex manifold and $\dim(X)=2$, $\eta$ is a holomorphic form on $X$. Prove that d$\eta=0$. I know when $X$ is a compact complex Kähler manifold, holomorphic forms are closed. In ...
save123's user avatar
  • 319
0 votes
0 answers
196 views

Chern class of a hypersurface in $\mathbb{C}P^3$

Let $X=\{[z_0,z_1,z_2,z_3]\ \big{|}\ [z_0,z_1,z_2,z_3]\in\mathbb{C}P^3,z_0^4+z_1^4+z_2^4+z_3^4=0\}$. $c_1(X)$ is the first Chern class of $X$. Prove that $c_1(X)=0$. $\textbf{My try}$: It's easy to ...
save123's user avatar
  • 319
0 votes
1 answer
95 views

affine Gauss map of Hypersurface Manifold finite

Let $M \subset \mathbb{C}^n $ be an algebraic $n-1$-dimensional manifold given as vanishing hypersurface of a polynomial $F \in \mathbb{C}[x_1,..., x_n]$ of degree $d \ge 2$. The smoothness of $M$ can ...
user267839's user avatar
  • 7,589
8 votes
1 answer
144 views

Is a finite covering of a $\partial\bar{\partial}$-manifold still $\partial\bar{\partial}$-manifold?

A compact complex manifold is called a $\partial\bar{\partial}$-manifold if for every pair $p,q\in \mathbb{N}$, every smooth $d$-closed $(p,q)$-form $\eta$ on $X$, $\eta$ is $d$-exact iff $\partial$-...
Doug's user avatar
  • 1,308
0 votes
0 answers
49 views

Decompose a finite (not necessarily positive) measure $\nu$ into two mutually singular measures $\lambda_1$ and $\lambda_2$

Let $\nu$ be a non-zero finite (not necessarily positive) measure on a compact Kähler surface $M$. Is it always possible to decompose the measure $\nu$ as follows $\nu=\lambda_1-\lambda_2$? Where $\...
Neil hawking's user avatar
  • 2,508
1 vote
2 answers
493 views

Is the Riemann Surface really a complex-manifold? Or is it without a point 0?

I just learnt basic complex analysis in school. My teacher simply described the Riemann Surface to explain the multi-valued functions better. I know that the so-called "multi-valued functions&...
HXR's user avatar
  • 41
1 vote
1 answer
62 views

the features of the action of $\Gamma = \langle \gamma \rangle$ on $\mathbb{C}\mathbb{P}^1$

Let $\gamma$ be an Elliptic element of ${\rm PSL}(2,\mathbb{C})$ representing an Irrational rotation. Let $\Gamma$ be the subgroup of ${\rm PSL}(2,\mathbb{C})$ generated by $\gamma$ (i.e., $\Gamma = \...
Neil hawking's user avatar
  • 2,508
1 vote
1 answer
233 views

Regarding the Defintion of Hirzebruch Surfaces $\mathbb{F}_n$

We have $\mathbb{F}_0= \mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ and, for $n \geqslant 1$, the $n-$th Hirzebruch Surfaces $\mathbb{F}_n$ is defined as a $\mathbb{C}\mathbb{P}^1$-bundle ...
Neil hawking's user avatar
  • 2,508
1 vote
0 answers
120 views

Finite dimensional space of vector fields

Consider a tangent bundle $TM$ and the space of sections $\mathfrak{X}(X)$. In general, for a $C^\infty$ structure, this space is an infinite dimensional vector space over $\mathbb{R}$. I'm trying to ...
BVquantization's user avatar
0 votes
0 answers
61 views

Let $z=a e^t$ then $dz=ae^tdt$, How to define $d \overline{z}=?$

Let $z$ and $t$ be two complex variables such that $z=a e^t$ where $a$ is a (real or complex) constant. Thus, the differential form $dz$ is nothing but $ae^tdt$, i.e., $$dz=ae^tdt.$$ The question is: ...
Neil hawking's user avatar
  • 2,508
2 votes
1 answer
80 views

Extension theorem in $\mathbb{C}^n$

Suppose $\Omega\subset \mathbb{C}^n$ is a nonempty open set and $V\subset \Omega$ is a complex analytic (closed) manifold. Let $f$ be a holomorphic function on $\Omega \backslash V$ and for every $p\...
user823011's user avatar
  • 1,345

1
2 3 4 5
7